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Theorem basis2 21563
 Description: Property of a basis. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
basis2 (((𝐵 ∈ TopBases ∧ 𝐶𝐵) ∧ (𝐷𝐵𝐴 ∈ (𝐶𝐷))) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷

Proof of Theorem basis2
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbasis2g 21560 . . . . 5 (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧))))
21ibi 270 . . . 4 (𝐵 ∈ TopBases → ∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)))
3 ineq1 4131 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝑧) = (𝐶𝑧))
4 sseq2 3941 . . . . . . . . . 10 ((𝑦𝑧) = (𝐶𝑧) → (𝑥 ⊆ (𝑦𝑧) ↔ 𝑥 ⊆ (𝐶𝑧)))
54anbi2d 631 . . . . . . . . 9 ((𝑦𝑧) = (𝐶𝑧) → ((𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
65rexbidv 3256 . . . . . . . 8 ((𝑦𝑧) = (𝐶𝑧) → (∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ ∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
76raleqbi1dv 3356 . . . . . . 7 ((𝑦𝑧) = (𝐶𝑧) → (∀𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ ∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
83, 7syl 17 . . . . . 6 (𝑦 = 𝐶 → (∀𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) ↔ ∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧))))
9 ineq2 4133 . . . . . . 7 (𝑧 = 𝐷 → (𝐶𝑧) = (𝐶𝐷))
10 sseq2 3941 . . . . . . . . . 10 ((𝐶𝑧) = (𝐶𝐷) → (𝑥 ⊆ (𝐶𝑧) ↔ 𝑥 ⊆ (𝐶𝐷)))
1110anbi2d 631 . . . . . . . . 9 ((𝐶𝑧) = (𝐶𝐷) → ((𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
1211rexbidv 3256 . . . . . . . 8 ((𝐶𝑧) = (𝐶𝐷) → (∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ ∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
1312raleqbi1dv 3356 . . . . . . 7 ((𝐶𝑧) = (𝐶𝐷) → (∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ ∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
149, 13syl 17 . . . . . 6 (𝑧 = 𝐷 → (∀𝑤 ∈ (𝐶𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝑧)) ↔ ∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
158, 14rspc2v 3581 . . . . 5 ((𝐶𝐵𝐷𝐵) → (∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) → ∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷))))
16 eleq1 2877 . . . . . . . 8 (𝑤 = 𝐴 → (𝑤𝑥𝐴𝑥))
1716anbi1d 632 . . . . . . 7 (𝑤 = 𝐴 → ((𝑤𝑥𝑥 ⊆ (𝐶𝐷)) ↔ (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))
1817rexbidv 3256 . . . . . 6 (𝑤 = 𝐴 → (∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷)) ↔ ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))
1918rspccv 3568 . . . . 5 (∀𝑤 ∈ (𝐶𝐷)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝐶𝐷)) → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))
2015, 19syl6com 37 . . . 4 (∀𝑦𝐵𝑧𝐵𝑤 ∈ (𝑦𝑧)∃𝑥𝐵 (𝑤𝑥𝑥 ⊆ (𝑦𝑧)) → ((𝐶𝐵𝐷𝐵) → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))))
212, 20syl 17 . . 3 (𝐵 ∈ TopBases → ((𝐶𝐵𝐷𝐵) → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))))
2221expd 419 . 2 (𝐵 ∈ TopBases → (𝐶𝐵 → (𝐷𝐵 → (𝐴 ∈ (𝐶𝐷) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷))))))
2322imp43 431 1 (((𝐵 ∈ TopBases ∧ 𝐶𝐵) ∧ (𝐷𝐵𝐴 ∈ (𝐶𝐷))) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3880   ⊆ wss 3881  TopBasesctb 21557 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499  df-uni 4801  df-bases 21558 This theorem is referenced by:  tgcl  21581  restbas  21770  txbas  22179  basqtop  22323  tgioo  23408
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